I really wouldn't worry about the fans at Fulham - mostly what we call 'plastic supporters' who weren't there 10 years ago when Fulham were a nothing team.
Mohammed (al) Fayed bought the club (as well as Harrods) and spent millions, encouraging new fans who were merely glory hunters or those who couldn't afford Chelsea prices. He still doesn't have a passport though (something my team like to remind him of...).
My advice would be to take the District line (green underground line) to Putney Bridge (NOT FULHAM BROADWAY) and take the short walk from there. There are a couple of pubs en-route.